Subgroups — C₉×#117 #63519

Order 288.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₉×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₉×C₂²⋊C₄	C₂	1	2	yes	maximal
4	C₄²×C₉	C₂	1	2	yes	maximal, complemented
5	C₃×#117	C₃	1	1	yes	maximal, characteristic
6	C₂³×C₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₉	C₄	1	4	yes	complemented
8	C₂×C₄×C₉	C₂²	1	4	yes	—
9	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄	C₂×C₃	1	1	yes	characteristic
12	C₃×C₄⋊C₄	C₂×C₃	1	2	yes	—
13	C₃×C₂²⋊C₄	C₂×C₃	1	2	yes	—
14	C₃×C₄²	C₂×C₃	1	2	yes	—
15	C₂²×C₉	—	2	2	no	—
16	C₂²×C₉	C₂×C₄	1	2	yes	—
17	C₂²×C₉	C₂³	1	1	yes	characteristic
18	C₄×C₉	—	2	4	no	—
19	C₄×C₉	C₂×C₄	1	2	yes	—
20	C₄×C₉	C₂×C₄	1	2	yes	—
21	#117	C₉	1	1	yes	characteristic, direct product factor, Sylow
22	C₂³×C₃	C₂²×C₃	1	1	yes	characteristic
23	C₂×C₃×C₄	C₃×C₄	1	4	yes	—
24	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
25	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
27	C₂×C₉	—	2	2	no	—
28	C₂×C₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₉	#41	1	2	yes	—
30	C₂²×C₄	C₂×C₉	1	1	yes	characteristic
31	C₄⋊C₄	C₂×C₉	1	2	yes	complemented
32	C₂²⋊C₄	C₂×C₉	1	2	yes	—
33	C₄²	C₂×C₉	1	2	yes	complemented
34	C₂²×C₃	—	2	2	no	—
35	C₂²×C₃	C₂×C₃×C₄	1	2	yes	—
36	C₂²×C₃	C₂³×C₃	1	1	yes	characteristic
37	C₃×C₄	—	2	4	no	—
38	C₃×C₄	C₂×C₃×C₄	1	2	yes	—
39	C₃×C₄	C₂×C₃×C₄	1	2	yes	—
40	C₉	#117	1	1	yes	characteristic, direct product factor, Sylow
41	C₂³	C₂²×C₉	1	1	yes	characteristic
42	C₂×C₄	C₄×C₉	1	4	yes	complemented
43	C₂×C₄	C₂²×C₉	1	4	yes	—
44	C₂×C₄	C₂²×C₉	1	1	yes	characteristic
45	C₂×C₄	C₂²×C₉	1	1	yes	characteristic
46	C₂×C₃	—	2	2	no	—
47	C₂×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
48	C₂×C₃	C₃×#41	1	2	yes	—
49	C₂²	—	2	2	no	—
50	C₂²	C₂×C₄×C₉	1	2	yes	—
51	C₂²	C₂³×C₉	1	1	yes	characteristic
52	C₄	—	2	4	no	—
53	C₄	C₂×C₄×C₉	1	2	yes	—
54	C₄	C₂×C₄×C₉	1	2	yes	—
55	C₃	C₃×#117	1	1	yes	characteristic
56	C₂	—	2	2	no	—
57	C₂	C₂²×C₄×C₉	1	1	yes	characteristic
58	C₂	C₉×#41	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₉	C₂²
commutator	C₂	C₂²×C₄×C₉
Frattini, socle, cosocle	C₂²×C₃	C₂³×C₃
prime radical	C₂³×C₃	C₂²×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₉×C₄⋊C₄	C₂	semidirect
4. C₄²×C₉	C₂	semidirect
7. C₂×C₄×C₉	C₄	semidirect
21. #117	C₉	direct (N × H)
31. C₄⋊C₄	C₂×C₉	semidirect
33. C₄²	C₂×C₉	semidirect
40. C₉	#117	direct (N × H)
42. C₂×C₄	C₄×C₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×C₉	2
C₉×C₄⋊C₄	2
C₉×C₂²⋊C₄	2
C₄²×C₉	2
C₃×#117	3

Subgroups — C9×#117 #63519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#117 #63519